Optimal. Leaf size=46 \[ \frac {(a+b) \tan ^3(c+d x)}{3 d}+\frac {a \tan (c+d x)}{d}+\frac {b \tan ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.04, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3675, 373} \[ \frac {(a+b) \tan ^3(c+d x)}{3 d}+\frac {a \tan (c+d x)}{d}+\frac {b \tan ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 373
Rule 3675
Rubi steps
\begin {align*} \int \sec ^4(c+d x) \left (a+b \tan ^2(c+d x)\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \left (1+x^2\right ) \left (a+b x^2\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a+(a+b) x^2+b x^4\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {a \tan (c+d x)}{d}+\frac {(a+b) \tan ^3(c+d x)}{3 d}+\frac {b \tan ^5(c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 53, normalized size = 1.15 \[ \frac {\tan (c+d x) \left (5 a \tan ^2(c+d x)+15 a+3 b \sec ^4(c+d x)-b \sec ^2(c+d x)-2 b\right )}{15 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 56, normalized size = 1.22 \[ \frac {{\left (2 \, {\left (5 \, a - b\right )} \cos \left (d x + c\right )^{4} + {\left (5 \, a - b\right )} \cos \left (d x + c\right )^{2} + 3 \, b\right )} \sin \left (d x + c\right )}{15 \, d \cos \left (d x + c\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.52, size = 48, normalized size = 1.04 \[ \frac {3 \, b \tan \left (d x + c\right )^{5} + 5 \, a \tan \left (d x + c\right )^{3} + 5 \, b \tan \left (d x + c\right )^{3} + 15 \, a \tan \left (d x + c\right )}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.59, size = 66, normalized size = 1.43 \[ \frac {b \left (\frac {\sin ^{3}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}\right )-a \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 39, normalized size = 0.85 \[ \frac {3 \, b \tan \left (d x + c\right )^{5} + 5 \, {\left (a + b\right )} \tan \left (d x + c\right )^{3} + 15 \, a \tan \left (d x + c\right )}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.94, size = 40, normalized size = 0.87 \[ \frac {\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^5}{5}+\left (\frac {a}{3}+\frac {b}{3}\right )\,{\mathrm {tan}\left (c+d\,x\right )}^3+a\,\mathrm {tan}\left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \tan ^{2}{\left (c + d x \right )}\right ) \sec ^{4}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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